Weak Convergence of Sums of Moving Averages in the $\alpha$-Stable Domain of Attraction

Abstract

Skorohod has shown that the convergence of sums of i.i.d. random variables to an $\alpha$-stable Levy motion, with $0 < \alpha < 2$, holds in the weak-$J_1$ sense. $J_1$ is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-$J_1$ convergence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-$M_1$ convergence. $M_1$ is weaker than $J_1$, but it is strong enough for the $\sup$ and $\inf$ functionals to be continuous.